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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

214 (2) The parameters

214 (2) The parameters ctj appearing in the Morse families in (ii) and (iii) are moduli. It follows from the next two remarks that these do not affect the diffeomorphism types of the corresponding caustics. (3) We show in the sequel to this paper, [JR], that the Morse families in (ii) have caustics which are ~'2 diffeomorphic to products of a smooth space with the causdc of the Morse family :- k ~2(k+l) + ~ yj~2j + Xl~. j=l with the same k. This caustic for k = 2, the symmetric biatterfly, is illustrated below. Note that for s = 2, r = 1 the Morse families:- 2. 6 + y2~. 4 + yl~. 2 + a(xl,Yl,Y2)Xl~. 3 + Xl~, where ~(xl,yl,y2) is any E2 invariant function germ, are not infinitesimally stable. O In fact TG(F) has infinite codimension in Epq. However these also have caustics diffeomorphic to the symmetric butterfly. These examples show that the caustic equivMence studied in [JR] is much weaker than the symplectic equivalence of this paper. The "symmetricbutterfly" caustic. See Remark 4.7 (3). \ \, f V

215 (4) In [JR] we show that the caustics of the Morse families in (iii) are equivariantly diffeomorphic to products of smooth spaces with the caustics of the families:- respectively. Proof of Theorem 4.5 3 ~8 + ~,yj~.2j + x2~3 + Xl~, j=l 3 ~8 + ~ yj~,2j + x2~5 + y4x2~ 3 + Xl~. ' j=l (i) This follows from Proposition 4.1 and Corollary 2.8. (ii) By Proposition 4.1, we may assume that F has the prenormal form:- k k F(~.,x,y) = ~.2(k+l) + ~ ya~.2a + ~ Vb(x,y)xl~2b_ 1 a=l b=l with ~1 = 1. By Remark 4.3, F is infinitesimally stable if and only if the matrix :- O(V 2 ........... V k .) (0,0) ~(Yk+l ......... Ys) has rank k- 1 (so, in particular k < t- 2 (s+l)). If this condition holds then we can change coordinates in V, leaving xl,y 1 ...... Yk unchanged, so that ~b(x,y) = Vb(0)+Yk+b_l. Setting aj = ~j+l(0) for j = 1 ..... k-1 gives the result. (iii) If s < 5 then by Corollary 4.4, F must be an unfolding of k2(k+l) for k < 2. The classification then follows from Corollary 2.8. If s -- 5 then k < 3. The cases k < 2 are again dealt with by Corollary 2.8. It remains to consider k = 3. Z 2 The ring, Epq , of 2 invariant function germs on W*@V is generated by:- A = ~,2, Zj = xj~,, Xij = xixj, Yl,Y2,Y3, Wl = Y4 and w 2 = YS- Z 2 The ring, Eq , of 2~ 2 invariant germs on V is generated by the Xij, Yi and w i. Let

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